• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1103-1112
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• 2018

We study the structure of a generalization of right nilpotent-duo rings in relation with powers of elements. Such a ring property is said to be weakly right nilpotent-duo. We find connections between weakly right nilpotent-duo and weakly right duo rings, in several algebraic situations which have roles in ring theory. We also observe properties of weakly right nilpotent-duo rings in relation with their subrings and extensions.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.177-191
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• 2017

In this note we consider the right unit-duo ring property on the powers of elements, and introduce the concept of weakly right unit-duo ring. We investigate first the properties of weakly right unit-duo rings which are useful to the study of related studies. We observe next various kinds of relations and examples of weakly right unit-duo rings which do roles in ring theory.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.3
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• pp.327-337
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• 2020

This article continues the study of right 𝜋-duo rings, concentrating on the situation of nonzero powers. For this purpose we introduce the concept of strongly right 𝜋-duo and examine the structure of strongly right 𝜋-duo in relation to various ring properties that play important roles in ring theory. It is proved for a strongly right 𝜋-duo ring R that if the upper (lower) nilradical of R is zero then R is reduced. Various kinds of examples are examined in relation to the questions raised in the procedure.

• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1233-1254
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• 2023

We introduce two subclasses of abelian McCoy rings, so-called π-CN-rings and π-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as 2-primal rings, bounded rings etc. It is shown that a ring R is π-CN whenever every nilpotent element of index 2 in R is central. These rings naturally generalize the long-known class of CN-rings, introduced by Drazin [9]. It is proved that π-CN-rings are abelian, McCoy and 2-primal. We also show that, π-duo rings are strongly McCoy and abelian and also they are strongly right AB. If R is π-duo, then R[x] has property (A). If R is π-duo and it is either right weakly continuous or every prime ideal of R is maximal, then R has property (A). A π-duo ring R is left perfect if and only if R contains no infinite set of orthogonal idempotents and every left R-module has a maximal submodule. Our achieved results substantially improve many existing results.